In 1, Koide's relation Q= (mₑ+m_μ+m_τ) / (√mₑ+√m_μ+√m_τ) ²=2/3 was derived as a consequence of the Z₃ symmetry of the chiral coupling operator with normalization condition |b|/a=1/√2. In the present work we investigate whether this symmetry extends to the neutrino sector. We prove that Q=2/3 is INCOMPATIBLE with observed neutrino masses in both orderings: in normal ordering (m₁<m₂<m₃), the maximum achievable value is Qₘax=0. 5829<2/3; in inverted ordering, the unique Q=2/3 solution predicts m₁=1644 meV with Σmᵢ=1702 meV, excluded by the cosmological bound Σmᵢ<120 meV. We show that the Z₃ circulant parametrization √mᵢ=A (1+√2·cos (θ+2πi/3) ) DOES apply to neutrinos, with parameters A_ν=3. 125 meV^½ and θ_ν=147. 7°, predicting m₁=0. 371 meV (Q_ν=0. 522). This prediction is falsifiable by PROJECT 8 (~2032) and future CMB surveys (CMB-S4). We analyze the connection between the Koide angle θ_ν and the CP violation phase in neutrinos δCP: θ_ν+θ₂₃=195. 3°≈δCP=195° (PDG). We discuss why the Z₃ symmetry works for charged leptons but gives Q≠2/3 for neutrinos, and what this implies for the flavor problem.
Danieletto et al. (Fri,) studied this question.